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Theorem grpideu 18180
 Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
Hypotheses
Ref Expression
grpcl.b 𝐵 = (Base‘𝐺)
grpcl.p + = (+g𝐺)
grpinvex.p 0 = (0g𝐺)
Assertion
Ref Expression
grpideu (𝐺 ∈ Grp → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Distinct variable groups:   𝑥,𝑢,𝐵   𝑢,𝐺,𝑥   𝑢, + ,𝑥   𝑥, 0
Allowed substitution hint:   0 (𝑢)

Proof of Theorem grpideu
StepHypRef Expression
1 grpmnd 18176 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2 grpcl.b . . 3 𝐵 = (Base‘𝐺)
3 grpcl.p . . 3 + = (+g𝐺)
42, 3mndideu 17988 . 2 (𝐺 ∈ Mnd → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
51, 4syl 17 1 (𝐺 ∈ Grp → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃!wreu 3072  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623  0gc0g 16771  Mndcmnd 17977  Grpcgrp 18169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-ov 7153  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172 This theorem is referenced by: (None)
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